Download Spatial Big Data: Case Studies on Volume, Velocity, and Variety

Abstract
Increasingly, the size, variety, and update rate of spatial datasets exceed the capacity of commonly
used spatial computing and spatial database technologies to learn, manage, and process data with
reasonable effort. We believe that this data, which we call Spatial Big Data (SBD), represents the next
frontier in spatial computing. Examples of emerging SBD include temporally detailed roadmaps that
provide traffic speed values every minute for every road in a city, GPS trajectory data from cell-phones,
and engine measurements of fuel consumption, greenhouse gas emissions, etc. A 2011 McKinsey Global
Institute report defines traditional big data as data featuring one or more of the 3 “V’s”: Volume, Velocity,
and Variety. This chapter discusses Spatial Big Data through case-studies on real-world datasets that
feature one or more of the 3 “V’s”: a study on change detection in climate data illustrates volume, a
study on finding anomalies in real-time highway traffic sensors shows Velocity, and two studies on Variety
demonstrate both variety in input and output. Spatial data has traditionally challenged traditional data
querying and mining algorithms, requiring new and interesting algorithms to be developed. Spatial Big
Data highlights these challenges and provides for a rich area of research.
1
Spatial Big Data: Case Studies on Volume, Velocity, and
Variety
Michael R. Evans
Dev Oliver
Xun Zhou
Shashi Shekhar
Department of Computer Science
University of Minnesota
Minneapolis, MN
1
Introduction
Spatial computing encompasses the ideas, solutions, tools, technologies, and systems that transform our
lives and society by creating a new understanding of spaces, their locations, places, as well as properties;
how we know, communicate, and visualize our relation to places in a space of interest; and how we navigate
through those places. From virtual globes to consumer global navigation satellite system devices, spatial
computing is transforming society. With the rise of new Spatial Big Data, spatial computing researchers
will be working to develop a compelling array of new geo-related capabilities. We believe that this data,
which we call Spatial Big Data (SBD), represents the next frontier in spatial computing. Examples of
emerging SBD include temporally detailed roadmaps that provide traffic speed values every minute for
every road in a city, GPS trajectory data from cell-phones, and engine measurements of fuel consumption,
greenhouse gas emissions, etc. A 2011 McKinsey Global Institute report defines traditional big data as
data featuring one or more of the 3 “V’s”: Volume, Velocity, and Variety [1]. Spatial data frequently
demonstrates at least one of these core features, given the variety of datatypes in spatial computing such
as points, lines, and polygons. In addition, spatial analytics have shown to be more computationally
expensive than their non-spatial brethren [2] as they need to account for spatial autocorrelation and
non-stationarity, etc.
In this chapter we begin in Section 2 by defining spatial big data, enumerating three traditional
categories of spatial data, and discussing their spatial big data equivalents. We then use case-studies to
demonstrate the 3 “V’s” of spatial big data: Volume, Velocity and Variety [1]. A case-study on climate
data in Section 3 illustrates the challenges of utilizing large volumes of spatial big data. Velocity is
demonstrated in Section 4 through a case-study on loop detector (traffic speed) data on the Twin Cities,
MN highway network. Lastly, variety in spatial big data can refer to both the type of data input used
and the variety in the type of output representations. We illustrate variety in data types through a
case-study on GPS trajectory data to find cyclist commuter corridors in Minneapolis, MN, and variety
in data output is demonstrated through network activity summarization of pedestrian fatality data from
Orlando, FL.
2
What is Spatial Big Data?
Spatial data are discrete representations of continuous phenomena. Discretization of continuous space
is necessitated by the nature of digital representation. There are three basic models to represent spatial
data: raster (grid), vector and network. Satellite images are good examples of raster data. On the
other hand, vector data consists of points, lines, polygons and their aggregate (or multi-) counterparts.
Graphs consisting of spatial networks are another important data type used to represent road networks.
We define Spatial Big Data as simply instances of these data types that exhibit at least one of the 3
“V’s”, Volume, Velocity and Variety. Below, we provide examples of Spatial Big Data in each of these
2
(a) Wide-area persistent surveillance.
FOV: Field of view. (Photo courtesy of
the Defense Advanced Research Projects
Agency.) EO: Electro-optical. [4]
(b) LIDAR images of ground zero
rendered Sept.
27, 2001 by the
U.S. Army Joint Precision Strike
Demonstration from data collected by
NOAA flights. Thanks to NOAA/U.S.
Army JPSD.
Figure 1: Spatial Big Data brings new challenges through data volume, velocity and variety.
core spatial data types: Raster, Vector, Network.
Raster data, such as geo-images (Google Earth), are frequently used for remote sensing and land
classification. New Spatial Big Raster Datasets are emerging from a number of sources:
UAV Data: Wide area motion imagery sensors are increasingly being used for persistent surveillance of
large areas, including densely populated urban areas. The wide-area video coverage and 24/7 persistent
surveillance of these sensor systems allow for new and interesting patterns to be found via temporal
aggregation of information. However, there are several challenges associated with using UAVs in gathering
and managing raster datasets. First, UAV has a small footprint due to the relatively low flying height,
therefore, it captures a large amount of images in a very short period of time to achieve the spatial coverage
for many applications. This poses a significant challenge to store increasing large digital images. Image
processing is another challenge because traditional approaches have shown to be too time consuming and
costly to rectify and mosaic the UAV photography for large areas. The large quantity of data far exceeds
the capacity of the available pool of human analysts [3]. It is essential to develop automated, efficient,
and accurate technique to handle these spatial big data.
LiDAR: Lidar (Light Detection and Ranging or Laser Imaging Detection and Ranging) data is
generated by timing laser pulses from an aerial position (plane or satellite) over a selected area to
produce a surface mapping [5]. Lidar data are very rich for use cases related to surface analysis or
feature extraction. However, these datasets are noisy and may contain irrelevant data for spatial analysis
and sometimes miss critical information. These large volumes of data from multiple sources pose a
big challenge on management, analysis, and timely accessibility. Particularly, Lidar points and their
attributes have tremendous sizes making them difficult to categorize these datasets for end-users. Data
integration from multiple spatial sources is another challenge due to the massive amounts of Lidar
datasets. Therefore, Spatial Big Data is an essential issue for Lidar remote sensing.
Vector data models over space is a framework to formalize specific relationships among a set of objects.
Vector data consists of points, lines and polygons; and with the rise of Spatial Big Data, corresponding
datasets have arisen from a variety of sources:
VGI Data: Volunteered geographic information (VGI) brings a new notion of infrastructure to collect,
synthesize, verify, and redistribute geographic data through geo-location technology, mobile devices, and
geo-databases. These geographic data are provided, modified, and shared based on user interactive online
services (e.g., OpenStreetMap, Wikimapia, GoogleMap, GoogleEarth, Microsofts Virtual Earth, Flickr,
etc). In recent years, VGI leads an explosive growth in the availability of user-generated geographic
information and requires scalable storage models to handle large-scale spatial datasets. The challenge
for VGI is to enhance data service quality with regard to accuracy, credibility, reliability, and overall
value [6].
GPS Trace Data: GPS trajectories are quickly becoming available for a larger collection of vehicles due
to rapid proliferation of cell-phones, in-vehicle navigation devices, and other GPS data-logging devices [7]
such as those distributed by insurance companies [8]. Such GPS traces allow indirect estimation of fuel
3
efficiency and greenhouse gas (GHG) emissions via estimation of vehicle-speed, idling and congestion.
They also make it possible to provide personalized route suggestions to users to reduce fuel consumption
and GHG emissions. For example, Figure 2 shows 3 months of GPS trace data from a commuter with
each point representing a GPS record taken at 1 minute intervals, 24 hours a day, 7 days a week. As can
be seen, 3 alternative commute routes were identified between home and work from this dataset. These
routes may be compared for engine idling which are represented by darker (red) circles. Assuming the
availability of a model to estimate fuel consumption from speed profiles, one may even rank alternative
routes for fuel efficiency. In recent years, consumer GPS products [7, 9] are evaluating the potential of
this approach. Again, a key hurdle is the dataset size, which can reach 1013 items per year given constant
minute-resolution measurements for all 100 million US vehicles.
(a) GPS trace data.
speed.
Color indicates
(b) Routes 1, 2, & 3 [10].
Figure 2: A commuter’s GPS tracks over three months reveal preferred routes. (Best viewed in color)
Network data is commonly used to represent road maps for routing queries. While the network
structure of the graph may not change, the amount of information about the network is rising drastically.
New temporally-detailed road maps give minute by minute speed information, along with elevation and
engine measurements to allow for more sophisticated querying of road networks.
Spatio-Temporal Engine Measurement Data: Many modern fleet vehicles include rich instrumentation
such as GPS receivers, sensors to periodically measure sub-system properties [11–16], and auxiliary
computing, storage and communication devices to log and transfer accumulated datasets. Engine
measurement datasets may be used to study the impacts of the environment (e.g., elevation changes,
weather), vehicles (e.g., weight, engine size, energy-source), traffic management systems (e.g., traffic
light timing policies), and driver behaviors (e.g., gentle acceleration or braking) on fuel savings and GHG
emissions. These datasets may include a time-series of attributes such as vehicle location, fuel levels,
vehicle speed, odometer values, engine speed in revolutions per minute (RPM), engine load, emissions of
greenhouse gases (e.g., CO2 and NOX), etc. Fuel efficiency can be estimated from fuel levels and distance
traveled as well as engine idling from engine RPM. These attributes may be compared with geographic
contexts such as elevation changes and traffic signal patterns to improve understanding of fuel efficiency
and GHG emission. For example, Figure 3 shows heavy truck fuel consumption as a function of elevation
from a recent study at Oak Ridge National Laboratory [17]. Notice how fuel consumption changes
drastically with elevation slope changes. Fleet owners have studied such datasets to fine-tune routes
to reduce unnecessary idling [18, 19]. It is tantalizing to explore the potential of such datasets to help
consumers gain similar fuel savings and GHG emission reduction. However, these datasets can grow big.
For example, measurements of 10 engine variables, once a minute, over the 100 million US vehicles in
existence [20, 21], may have 1014 data-items per year.
Historical Speed Profiles: Typically, digital road maps consist of center lines and topologies of road
networks [22, 23]. These maps are used by navigation devices and web applications such as Google
Maps [10] to suggest routes to users. New datasets from companies such as NAVTEQ [24], use probe
vehicles and highway sensors (e.g., loop detectors) to compile travel time information across road segments
throughout the day and week at fine temporal resolutions (seconds or minutes). This data is applied to
a profile model, and patterns in the road speeds are identified throughout the day. The profiles have
4
Figure 3: Engine measurement data improve understanding of fuel consumption [17]. (Best in color)
data for every five minutes, which can then be applied to the road segment, building up an accurate
picture of speeds based on historical data. Such TD roadmaps contain much more speed information
than traditional roadmaps. While traditional roadmaps have only one scalar value of speed for a given
road segment (e.g., EID 1), TD roadmaps may potentially list speed/travel time for a road segment
(e.g., EID 1) for thousands of time points, see Figure 4(a), in a typical week. This allows a commuter
to compare alternate start-times in addition to alternative routes. It may even allow comparison of
(start-time, route) combinations to select distinct preferred routes and distinct start-times. For example,
route ranking may differ across rush hour and non-rush hour and in general across different start times.
However, TD roadmaps are large and their size may exceed 1013 items per year for the 100 million
road-segments in the US when associated with per-minute values for speed or travel-time. Thus, industry
is using speed-profiles, a lossy compression based on the idea of a typical day of a week, as illustrated in
Figure 4(b), where each (road-segment, day of the week) pair is associated with a time-series of speed
values for each hour of the day.
(a) Travel time along four road
segments over a day.
(b) Schema for Daily Historic Speed Data.
Figure 4: Spatial Big Data on Historical Speed Profiles. (Best viewed in color)
3
Volume: Discovering Sub-paths in Climate Data
Sub-paths (i.e., intervals) in spatio-temporal (ST) datasets can be defined as contiguous subsets of
locations. Given a ST dataset and a path in its embedding ST framework, the goal of the interesting
spatio-temporal sub-path discovery problem is to identify all the dominant (i.e., not a subset of any
other) interesting sub-paths along the path defined by a given interest measure. The ability to discover
interesting sub-paths is important to many societal applications. For example, coastal area authorities
may be interested in intervals of coastal lines which are prone to rapid environmental change due to rising
ocean levels and melting polar icecaps. Water quality monitors may be interested in river segments where
water quality changes abruptly.
An extended example from eco-climate science illustrates the interesting sub-path discovery problem
in details. This example comes from our collaboration with scientists studying the response of ecosystems
to climate change by observing changes in vegetation cover across ecological zones. Sub-paths of abrupt
vegetation cover change may serve to outline the spatial footprint of ecotones, the transitional areas
between these zones [25]. Due to their vulnerability to climate changes, finding and tracking ecotones
gives us important information about how the ecosystem responds to climate changes. Figure 5 illustrates
the application of interesting sub-path discovery on the Africa vegetation cover in normalized difference
vegetation index (NDVI) data, August, 1981. Figure 5(a) shows a map of vegetation cover in Africa [26].
Each longitudinal path is taken as an input of the problem. The output, as shown in Figure 5 (b), is
a map of longitudinal sub-paths with abrupt vegetation cover changes, where red and blue represent
sub-paths of abrupt vegetation cover decrease and increase northward respectively. As indicated by the
5
43.7
43.7
0.9
0.8
0.7
22.2
22.2
0.5
0.7
0.4
Latitude
Latitude
0.6
0.7
0.3
0.2
−20.7
−20.7
0.1
Abrupt increase
0
Abrupt decrease
−42.2
−24.6
−2.6
19.4
41.4
64.4
−0.1
Longitude
−42.2
−24.6
−2.6
19.4
41.4
64.4
Longitude
(a) Smoothed Africa vegetation dataset (b) Longitudinal intervals of abrupt
(measured in NDVI) in August, 1981.
vegetation change in August, 1981
Figure 5: An application example of the interesting interval discovery problem. (Best viewed in color)
two colors as in Figure 5, footprints of several ecotones in Africa are discovered. One of them is the Sahel
region (in the middle in red), where vegetation cover exhibits an abrupt decreasing trend from south to
north.
Discovering interesting sub-paths is challenging due to the following reasons. First, the length of
the sub-paths of interest may vary, without a pre-defined maximum length. For example, the length of
flood-prone interval in long rivers (e.g., the Gange, Mississippi, etc.) may extend hundreds or thousands of
miles. Second, the interestingness in a sub-path may not exhibit monotonicity, i.e., uninteresting intervals
may be included in an interesting sub-path. Third, the data volume is potentially large. For example,
consider the problem of finding all the interesting longitude sub-paths exhibiting abrupt change in an
eco-climate dataset with attributes such as vegetation, temperature, precipitation, etc., over hundreds
of years from different global climate models and sensor networks. The volume of such spatial big data
ranges from terabytes to petabytes.
Previous work on interesting spatiotemporal sub-path/interval discovery focused on change point
detection using one dimensional or two dimensional approaches. One dimensional approaches aim to
find points in a time series where there is a shift in the data distribution [27–29]. Figure 6(a) shows a
sample dataset in vegetation cover along a particular longitude. Figure 6(b) shows an sub-path in this
dataset from location 5 to 11 whose data exhibits an abruptly increasing trend. In contrast, Figure 6(c)
shows the output of a specific implementation of the popular statistical measure CUSUM [27, 30] on
the same data where only location 6 is identified as a point of interest (with abrupt change from below
the mean to above the mean). Two dimensional approaches such as edge detection [31] aim at finding
boundaries between different areas in an image. However, the footprints of an identified edge over each
one-dimensional path (e.g., row or column) are still points. The above related works are limited to
detecting points of interest in a ST path, rather than finding long interesting sub-paths/intervals. In
contrast, our novel computational frameworks discover sub-paths of arbitrary length based on certain
interest measures.
In our preliminary work [32], a sub-path enumeration and pruning (SEP) approach was proposed.
In the approach, the enumeration space of all the sub-paths is modeled as a grid-based directed acyclic
graph (G-DAG), where nodes are sub-paths and edges are subset relationships between sub-paths. The
approach enumerates all the sub-path by performing a breadth-first traversal on the G-DAG, starting
from the root (longest sub-path). Each sub-path is evaluated by commuting its algebraic interest measure.
Should an interesting sub-path be identified, all its subsets are pruned. By doing this, we significantly
reduce the number of sub-path evaluations. We apply this approach on eco-climate datasets to find
abrupt change sub-paths. An algebraic interest measure named ”sameness degree” was designed to
evaluate the change abruptness and persistence of a sub-path. As noted earlier, case study results on
Normalized Difference Vegetation Index (NDVI) vegetation cover dataset showed that the approach can
discover important patterns such as ecotones (e.g., the Sahel region). We also apply this approach on
temporal paths (e.g., precipitation time series) and discovered patterns such as abrupt precipitation
shifts in Africa. Experimental results on large synthetic datasets confirmed that the proposed approach
is efficient and scalable.
6
(a) Vegetation cover
longitudinal path
along
a (b) An interesting sub-path in the
path
(c) Result of CUSUM
Figure 6: A comparison of interesting sub-paths in the data and change point found by related work. (Best
viewed in color)
(a) Traffic speed detector
map.
(b) Spatio-temporal outlier detection framework
Figure 7: Detecting outliers in real-time traffic data.
4
Velocity: Spatial Graph Outlier Detection in Traffic Data
In this section, we demonstrate spatial big data featuring velocity via a case-study on real-time traffic
monitoring datasets for detecting outliers in spatial graph datasets. We formalize the problem of
spatio-temporal outlier detection and propose an efficient graph-based outlier detection algorithm. We
use our algorithm to detect spatial and temporal outliers in a real-world Minneapolis-St. Paul dataset,
and show effectiveness of our approach.
In 1997, the University of Minnesota and the Track Management Center Freeway Operations group
started a joint project to archive sensor network measurements from the freeway system in the Twin
Cities [33]. The sensor network includes about nine hundred stations, each of which contains one to
four loop detectors, depending on the number of lanes. Sensors embedded in the freeways and interstate
monitor the occupancy and volume of track on the road. At regular intervals, this information is sent
to the track Management Center for operational purposes, e.g., ramp meter control, as well as research
on track modeling and experiments. Figure 7(a) shows a map of the stations on the highways within
the Twin-Cities metropolitan area, where each polygon represents one station. The interstate freeways
include I-35W, I35E, I-94, I-394, I-494, and I-694. The state trunk highways include TH-100, TH-169,
TH-212, TH-252, TH-5, TH-55, TH-62, TH-65, and TH-77. I-494 and I-694 together form a ring around
the Twin-Cities. I-94 passes from East to North-West, while I-35W and I-35E run in a South-North
direction. Downtown Minneapolis is located at the intersection of I-94, I-394, and I-35W, and downtown
Saint Paul is located at the intersection of I-35E and I-94. For each station, there is one detector installed
in each lane. The track flow information measured by each detector can then be aggregated to the station
level. The system records all the volume and occupancy information within each 5-minute time slot at
each particular station.
In this application, we are interested in discovering 1) the location of stations whose measurements
are inconsistent with those of their graph-based spatial neighbors and 2) time periods when those
7
(a) Station 138
(b) Station 139
(c) Station 140
Figure 8: Outlier station 139 and its neighbor stations over one day.
abnormalities arise. We use three neighborhood definitions in this application as shown in Figure 7(b).
First, we define a neighborhood based on the spatial graph connectivity as a spatial graph neighborhood.
In Figure 7(b), (s1; t2) and (s3; t2) are the spatial neighbors of (s2; t2) if s1 and s3 are connected to s2 in
a spatial graph. Second, we define a neighborhood based on a time series as a temporal neighborhood. In
Figure 7(b), (s2; t1) and (s2; t3) are the temporal neighbors of (s2; t2) if t1, t2, and t3 are consecutive time
slots. In addition, we define a neighborhood based on both space and time series as a spatial-temporal
neighborhood. In Figure 7(b), (s1; t1), (s1; t2), (s1; t3), (s2; t1), (s2; t3), (s3; t1), (s3; t2), and (s3; t3)
are the spatial-temporal neighbors of (s2; t2) if s1 and s3 are connected to s2 in a spatial graph, and t1,
t2, and t3 are consecutive time slots.
s
The test for detecting an outlier can be described as follows: | S(x)−µ
| > θ For each data object x
σs
with an attribute value f (x), the S(x) is the difference of the attribute value of data object x and the
average attribute value of its neighbors. µs is the mean value of all S(x), and σs is the standard deviation
of all S(x). Choice of θ depends on specified confidence interval. For example, a confidence interval of
95 percent will lead to θ ≈ 2.
In prior work [33], we proposed an I/O efficient algorithm to calculate the test parameters, e.g., mean
and standard deviation for the statistics. The computed mean and standard deviation can then be used
to validate the outlier of the incoming dataset. Given an attribute dataset V and the connectivity graph
G, the TPC algorithm first retrieves the neighbor nodes from G for each data object x, then it computes
the difference of the attribute value of x to the average of the attribute values of x’s neighbor nodes.
These different values are then stored as a set. Finally, that set is computed to get the distribution value
µs and σs . Note that the data objects are processed on a page basis to reduce redundant I/O. In other
words, all the nodes within the same disk page are processed before retrieving the nodes of the next disk
page.
The neighborhood aggregate statistics value, e.g., mean and standard deviation, can be used to verify
the outlier of an incoming dataset. The two verification procedures are Route Outlier Detection and
Random Node Verification. The Route Outlier Detection procedure detects the spatial outliers from
a user specified route. The Random Node Verification procedure check the outlierness from a set of
randomly generated nodes. The step to detect outliers in both algorithms are similar, except that the
Random Node Verification has no shared data access needs across test for different nodes. The storage
of dataset should support I/O efficient computation of this operation.
Given a route RN in the dataset D with graph structure G, the Route Outlier Detection algorithm
first retrieves the neighboring nodes form G for each data object x in the route RN , then it computes the
difference S(x) between the attribute value of x and the average of attribute values of x’s neighboring
s
nodes. Each S(x) can then be tested using the spatial outlier detection test | S(x)−µ
| > θ. The θ is
σs
predetermined by the given confidence interval.
We tested the effectiveness of our algorithm on the Twin-Cities track dataset and detected numerous
outliers, as described in the following examples. In Figure 8(b), the abnormal station (Station 139)
was detected whose volume values are significantly inconsistent with the volume values of its neighboring
stations 138 and 140. Note that our basic algorithm detects outlier stations in each time slot; the detected
outlier stations in each time slot are then aggregated to a daily basis.
Figure 8 shows an example of loop detector outliers. Figures 8(a) and 8(c) are the track volume maps
for I-35W North Bound and South Bound, respectively, on 1/21/1997. The X-axis is the 5-minute time
slot for the whole day and the Y-axis is the label of the stations installed on the highway, starting from 1
8
(a) Recorded GPS points from bicyclists (b) Set of 8 -Primary Corridors identified
in Minneapolis, MN. Intensity indicates from Bike GPS traces in Minneapolis, MN.
number of points.
(Best in color)
Figure 9: Example input and output of the k-Primary Corridor problem.
in the north end to 61 in the south end. The abnormal dark line at time slot 177 and the dark rectangle
during time slot 100 to 120 on X-axis and between station 29 to 34 on Y-axis can be easily observed from
both 8(a) and 8(c). This dark line at time slot 177 is an instance of temporal outliers, where the dark
rectangle is a spatial-temporal outlier. Moreover, station 9 in Figure 8(a) exhibits inconsistent track data
compared with its neighboring stations, and was detected as a spatial outlier.
5
Variety in Data Types: Identifying Bike Corridors
Given a set of trajectories on a road network, the goal of the k-Primary Corridors problem is to summarize
trajectories into k groups, each represented by its most central trajectory. Figure 10(a) shows a real-world
GPS dataset of a number of trips taken by bicyclists in Minneapolis, MN. The darkness indicates the
usage levels of each road segment. The computational problem is summarizing this set of trajectories
into a set of k-Primary Corridors. One potential solution is shown in Figure 9(b) for k=8. Each identified
primary corridor represents a subset of bike tracks from the original dataset. Note that the output of
the k-PC problem is distinct from that of hot or frequent routes, as it is a summary of all the given
trajectories partitioned into k primary corridors.
The k-Primary Corridor (kPC) problem is important due to a number of societal applications,
such as city-wide bus route modification or bicycle corridor selection, among other urban development
applications. Let us consider the problem of determining primary bicycle corridors through a city to
facilitate safe and efficient bicycle travel. By selecting representative trajectories for a given group
of commuters, the overall alteration to commuters routes is minimized, encouraging use. Facilitating
commuter bicycle traffic has shown in the past to have numerous societal benefits, such as reduced
greenhouse gas emissions and healthcare costs [34].
Clustering trajectories on road networks is challenging due to the computational cost of computing
pairwise graph-based minimum-node-distance similarity metrics between trajectories in large GPS datasets
as shown in our previous work [35]. We proposed a baseline algorithm using a graph-based approach to
compute a single element of the trajectory similarity matrix, requiring multiple invocations of common
shortest-path algorithms (e.g., Dijkstra [36]). For example, given two trajectories consisting of 100 nodes
each, a baseline approach to calculate NHD would need to compute the shortest distance between all
pairs of nodes (104 ), which over a large trajectory dataset (e.g., 10,000 trajectories) would require 1012
shortest path distance computations. This quickly becomes computationally prohibitive without faster
algorithms.
Trajectory pattern mining is a popular field with a number of interesting problems both in geometric
(Euclidean) spaces [37] and networks (graphs) [38]. A key component to traditional data mining in these
domains is the notion of a similarity metric, the measure of sameness or closeness between a pair of
objects. A variety of trajectory similarity metrics, both geometric and network, have been proposed in
the literature [39]. One popular metric is Hausdorff distance, a commonly used measure to compare
similarity between two geometric objects (e.g., polygons, lines, sets of points) [40]. A number of methods
have focused on applying Hausdorff distance to trajectories in geometric space [41–44].
We formalize the Network Hausdorff Distance and propose a novel approach that is orders of magnitude
faster than the baseline approach and our previous work, allowing for Network Hausdorff Distance to
9
be computed efficiently on large trajectory datasets. A baseline approach for solving the kPC problem
would involve comparing all nodes in each pairwise combination of trajectories. That is, when comparing
two trajectories, each node in the first trajectory needs to find the shortest distance to any node in the
opposite trajectory. The maximum value found is the Hausdorff distance [40]. While this approach does
find the correct Hausdorff distance between the two trajectories, it is computationally expensive due to
the node-to-node pairwise distance computations between each pair of trajectories. We will demonstrate
using experimental and analytical analysis how prohibitive that cost is on large datasets. Due to this,
related work solving the kPC problem has resulted in various heuristics [45–49].
We propose a novel approach that ensures correctness while remaining computationally efficient.
While the baseline approach depends on computing node-to-node distances, the Network Hausdorff
Distance (NHD) requires node-to-trajectory minimum distance. We take advantage of this insight to
compute the NHD between nodes and trajectories directly by modifying the underlying graph and
computing from a trajectory to an entire set of trajectories with a single shortest-paths distance computation.
This approach retains correctness while proving significantly faster than the baseline approach and our
previous work [35].
Our previous work [35] in the kPC problem requires O(|T |2 ) invocations of a shortest-path algorithm
to compute the necessary trajectory similarity matrix (TSM), becoming prohibitively expensive when
dealing with datasets with a large number of trajectories. Therefore, we developed a novel row-wise
algorithm to compute the kPC problem on spatial big data.
Computing the Network Hausdorff Distance N HD(tx , ty ) between two trajectories does not require
the shortest distance between all-pairs of nodes in tx and ty . We require the shortest network distance
from each node in tx to the closest node in ty . In [35], we proposed a novel approach to find this distance,
as compared to enumerating the all-pair shortest-path distances as GNTS does. This significantly reduced
the number of distance calculations and node iterations needed to compute the TSM. In Figure 10(b), to
calculate N HD(tB , tA ), we began by inserting a virtual node (Avirtual ) representing Trajectory tA into
the graph. This node had edges with weights of 0 connecting it to each other node in Trajectory tA . We
then ran a shortest-path distance computation from the virtual node as a source, with the destination
being every node in Trajectory tB . The result was the shortest distance from each node in Trajectory
tB to the virtual node Avirtual . Since the virtual node is only connected to nodes in Trajectory tA , and
all the edge weights are 0, we had the shortest-path from each node in Trajectory tB to the closest node
in Trajectory tA , exactly what N HD(tB , tA ) requires for computation.
However, our previous work [35] focused on computing a single cell in the trajectory similarity matrix
per invocation of a single-source shortest-paths algorithm. That meant at least O(|T |2 ) shortest-path
invocations to compute the TSM for the kPC problem, still quite expensive. We propose ROW-TS to
compute an entire row of the TSM with one invocation of a single-source shortest-paths algorithm. Using
a row-based approach, we can essentially calculate N HD(t ∈ T, tA ) with one single-source shortest-paths
invocation from Avirtual . This approach reduces the overall number of shortest-paths invocations to
O(|T |) at the cost of additional bookkeeping, as we will show below. However, due to the expensive cost
of shortest-path algorithms, this results in significant performance savings.
In Summer 2006, University of Minnesota researchers collected a variety of data to help gain a better
understanding of commuter bicyclist behavior using GPS equipment and personal surveys to record
bicyclist movements and behaviors [50]. The broad study examined a number of interesting issues with
commuter cycling, for example, results showed that as perceived safety decreases (possibly due to nearby
vehicles), riders appear to be more cautious and move more slowly. One possible issue they looked at was
identifying popular transportation corridors for the Minnesota Department of Transportation to focus
funds and repairs. At the time, they hand-crafted the primary corridors. Shortly after this study, the
U.S. Department of Transportation began a four-year, $100 million pilot project in four communities
(including Minneapolis) aimed to determine whether investing in bike and pedestrian infrastructure
encouraged significant increases in public use. As a direct result of this project, the US DoT found
that biking increased 50%, 7,700 fewer tons of carbon dioxide were emitted, 1.2 fewer gallons of gas was
burned, and there was a $6.9 million/year reduction in health care costs [34].
6
Variety in Output: Spatial Network Activity Summarization
Spatial network activity summarization (SNAS) is important in several application domains including
crime analysis and disaster response [51]. For example, crime analysts look for concentrations of individual
events that might indicate a series of related crimes [52]. Crime analysts need to summarize such incidents
on a map, so that law enforcement is better equipped to make resource allocation decisions [52]. In disaster
10
(a) Road network represented as an
undirected graph with four trajectories
illustrated with bold dashed lines.
(b) Inserting a virtual node (Avirtual ) to represent
Track A for efficient Network Hausdorff Distance
computation.
Figure 10: By altering the underlying graph, we can quickly compute the Network Hausdorff Distance.
response-related applications, action is taken immediately after a disastrous event with the aim of saving
life, protecting property, and dealing with immediate disruption, damage or other effects caused by the
disaster [53]. Disaster response played an important role in the 2010 earthquake in Haiti, where there
were many requests for assistance such as food, water and medical supplies [54]. Emergency managers
need the means to summarize these requests efficiently so that they can better understand how to allocate
relief supplies.
The SNAS problem is defined informally as follows: Given a spatial network, a collection of activities
and their locations (e.g., a node or an edge), a set of paths P , and a desired number of paths k, find a
subset of k paths in P that maximizes the number of activities on each path. An activity is an object of
interest in the network. In crime analysis, an activity could be the location of a crime (e.g., theft). In
disaster response-related applications, an activity might be the location of a request for relief supplies.
Figures 10(a) and 9(b) illustrate an input and output example of SNAS, respectively. The input consists
of six nodes, eight edges (with edge weights of 1 for simplicity), fourteen activities, the set of shortest
paths for this network, and k = 2, indicating that two routes are desired. The output contains two routes
from the given set of shortest paths that maximize the activity coverage; route hC, A, Bi covers activities
1, 2, 3, 4, 5 and route hC, D, Ei covers activities 7, 8, 12, 13, 14.
In network-based summarization, spatial objects are grouped using network (e.g., road) distance.
Existing methods of network-based summarization such as Mean Streets [55], Maximal Subgraph Finding
(MSGF) [56], and Clumping [57–64] group activities over multiple paths, a single path/subgraph or no
paths at all. Mean Streets [55] finds anomalous streets or routes with unusually high activity levels. It is
not designed to summarize activities over k paths because the number of high crime streets returned is
always relatively small. MSGF [56] identifies the maximal subgraph (e.g., a single path, k = 1) under the
constraint of a user specified length and cannot summarize activities when k > 1. The Network-Based
Variable-Distance Clumping Method (NT-VCM) [64] is an example of the clumping technique [57–64].
NT-VCM groups activities that are within a certain shortest path distance of each other on the network;
in order to run NT-VCM, a distance threshold is needed.
In this chapter, we propose a K-Main Routes (KMR) approach that finds a set of k routes to summarize
activities. KMR aims to maximize the number of activities covered on each k route. KMR employs an
inactive node pruning algorithm where instead of calculating the shortest paths between all pair of nodes,
only shortest paths between active nodes and all other nodes in the spatial network are calculated. This
results in computational savings (without affecting the resulting summary paths) that are reported in
the experimental evaluation. The inputs of KMR include the following: 1) an undirected spatial network
G = (N, E), 2) a set of activities A and 3) a number of routes, k, where k ≥ 1. The output of KMR is a
set of k routes where the objective is to maximize the activity coverage of each k route. Each k route is
a shortest path between its end-nodes and each activity ai ∈ A is associated with only one edge ei ∈ E.
KMR first calculates shortest paths between active nodes and all other nodes and then selects k
shortest paths as initial summary paths. The main loop of KMR assigns and updates steps of phases 1
and 2 are repeated until the summary paths do not change.
Phase 1: Assign activities to summary paths. The first step of this phase initializes the set of next
clusters, i.e., nextClusters, to the empty set. In general, this phase is concerned with forming k clusters
by assigning each activity to its nearest summary path. To accomplish this, KMR considers each activity
and each cluster in determining the nearest summary path to an activity.
KMR uses the following proximity measure to quantify nearness: prox(si , ai ). The distance between
an activity and a summary path is the minimum network distance between each node of the edge that
11
(a) Input
(b) KMR
(c) Crimestat K-means with Euclidean Distance (d) Crimestat K-means with Network Distance
Figure 11: Comparing KMR and Crimestat K-means output for k = 4 on pedestrian fatality data from
Orlando, FL [65].
the activity is on and each node of the summary path. Once the distance from each activity to each
summary path is calculated, the activity is assigned to the cluster with the nearest summary path.
Phase 2: Recompute summary paths. This phase is concerned with recomputing the summary path of
each cluster so as to further maximize the activity coverage. This entails iterating over each cluster and
initializing the summary path for each cluster. The summary path of each cluster is updated based on
the activities assigned to the cluster. The summary path with the maximum activity coverage is chosen
as the new summary path for each cluster ci , i.e., spmax ← max(AC(spk ) ∈ ci | k = 1...|sp|). Phases 1
and 2 are repeated until the summary paths of each cluster do not change. At the end of each iteration
the current clusters are initialized to the next clusters. Once the summary paths of each cluster do not
change, a set of k routes is returned.
We conducted a qualitative evaluation of KMR comparing its output with the output of Crimestat [66]
K-means [67] (a popular summarization technique) on a real pedestrian fatality dataset [65], shown in
Figure 11(a). The input consists of 43 pedestrian Fatalities (represented as dots) in Orlando, Florida
occurring between 2000 and 2009. As we have explained, KMR uses paths and network distance to
group activities on a spatial network. By contrast, in geometry-based summarization, the partitioning
of spatial data is based on grouping similar points distributed in planar space where the distance is
calculated using Euclidean distance. Such techniques focus on the discovery of the geometry (e.g., circle,
ellipse) of high density regions [52] and include K-means [?, 67–71], K-medoid [72, 73], P-median [74] and
Nearest Neighbor Hierarchical Clustering [75] algorithms.
Figures 11(b), 11(c), and 11(d) show the results of KMR, K-means using Euclidean distance, and
K-Means using network distance, respectively. In all cases, K was set to 4. The output of each technique
shows (1) the partitioning of activities represented by different colors and (2) the representative of each
partition (e.g., paths or ellipses). For example, Figure 11(c) shows (1) activities that are colored brown,
red, green, and blue, representing the four different partitions to which each activity belongs and (2)
ellipses representing each partition of activities (e.g., the red ellipse is the representative for the red
partition of activities).
This work explored the problem of spatial network activity summarization in relation to important
application domains such as crime analysis and disaster response. We proposed a K-Main Routes (KMR)
algorithm that discovers a set of k paths to group activities. KMR uses inactive node pruning, Network
Voronoi activity Assignment and Divide and Conquer Summary Path Recomputation to enhance its
performance and scalability. Experimental evaluation using both synthetic and real-world datasets
12
indicated that the performance-tuning decisions utilized by KMR yielded substantial computational
savings without reducing the coverage of the resulting summary paths. For qualitative evaluation, a
case study comparing the output of KMR with the output of a current geometry-based summarization
technique highlighted the potential usefulness of KMR to summarize activities on spatial networks.
7
Summary
Increasingly, location-aware datasets are of a size, variety, and update rate that exceed the capability
of spatial computing technologies. This chapter discussed some of the emerging challenges posed by
such datasets, referred to as Spatial Big Data (SBD). SBD examples include trajectories of cell-phones
and GPS devices, temporally detailed (TD) road maps, vehicle engine measurements, etc. SBD has the
potential to transform society. A recent McKinsey Global Institute report estimates that spatial big
data, such as personal location data, could save consumers hundreds of billions of dollars annually by
2020 by helping vehicles avoid congestion via next-generation routing services such as eco-routing.
Spatial Big Data has immense potential to benefit a number of societal applications. By harnessing
this increasingly large, varied and changing data, new opportunities to solve worldwide problems are
presented. To capitalize on these new datasets, inherent challenges that come with spatial big data
need to be addressed. For example, many spatial operations are iterative by nature, something that
parallelization has not yet been able to handle completely. By expanding cyber-infrastructure, we can
harness the power of these massive spatial datasets. New forms of analytics using simpler models and
richer neighborhoods will enable solutions in a variety of disciplines.
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